This is a myth that's sprung up. The 'higher pressures' measured in 5.56 is the same difference you could get changing primers or firing a load in hot and cold temps.

Military 5.56 rifles have a longer lead (aka freebore) to allow for 62 grain and upward weight projectiles so when you use this ammunition in a 223 (of which many have a shorter leade) the danger is that the bullets gets jammed in the lands and dangerous pressures occur. In practice however i've never heard of this happening and when you consider that some people jam bullets into the lands on purpose (I'm looking at benchrest guys here) you've got to wonder about this as well (im not recommending it though - i don't do it myself).

All civilian arms must also pass SAAMI or CIP proof tests which is far higher then normal operating pressures.

To cap it off i've tested 5.56mm military surplus ammo in my 223 and it worked perfectly fine. What you've done it fine. Load it up

(and just out of interest, if the 5.56mm was ore powerful then how would shooting a lower pressure round hurt it?)